Move Render Region With Animation
It's not possible to have the render region track an object by default, however, I did make an add-on for this exact purpose: 'Animated Render Border',
available on the Blender Market here.

(Gif is from 2.79, but the add-on works with 2.8+.)
The add-on can track an object by looking at the object's bounding box, or more precisely, using its vertices (if it is a mesh, curve, armature (uses bones) or lattice). It can track collections of objects and it also has a mode to allow a more manual fine tuning ('Keyframe' mode).